Koopman Kernel Regression

TL;DR

This paper introduces Koopman Kernel Regression, a new framework that leverages RKHS to provide theoretical guarantees and improved forecasting of complex dynamical systems in decision-making tasks.

Contribution

It develops a universal Koopman-invariant RKHS enabling statistically grounded learning guarantees for Koopman-based models.

Findings

Superior forecasting accuracy over existing Koopman and sequential predictors

Provides convergence results and generalization bounds under weaker assumptions

Demonstrates practical effectiveness in complex dynamical systems

Abstract

Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix multiplication. Though there exists a variety of learning approaches, they usually lack crucial learning-theoretic guarantees, making the behavior of the obtained models with increasing data and dimensionality unclear. We address the aforementioned by deriving a universal Koopman-invariant reproducing kernel Hilbert space (RKHS) that solely spans transformations into LTI dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization error bounds under weaker assumptions than existing work. Our experiments demonstrate superior forecasting performance compared to Koopman operator and sequential data predictors in RKHS.